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Supplementary Materials:
Broadband perfect light trapping in the thinnest monolayer
graphene-MoS2 photovoltaic cell: the new application of
spectrum-splitting structure
Yun-Ben Wu1, Wen Yang2, Tong-Biao Wang3, Xin-Hua Deng3, Jiang-Tao Liu1,3*
1Nanoscale
Science and Technology Laboratory, Institute for Advanced Study, Nanchang
University, Nanchang 330031, China
2Beijing Computational Science Research Center, Beijing 100084, China
3Department of Physics, Nanchang University, Nanchang 330031, China
Corresponding Author Email: [email protected]
The supplement includes more details on:
S1. Transfer matrix method.
S2. Distributed Bragg reflector with parallel layers.
S3. The influence of the fabrication tolerances of the photonic crystal.
S4. The outermost layer correction.
S5. The tolerances of the photovoltaic cells position.
S6. The external quantum efficiency of the proposed GM-PV cell.
S1. Transfer matrix method.
The modified transfer matrix method is used to model the absorption of monolayer
graphene-MoS2 photovoltaic cell (GM-PV cell) in the photonic crystal microcavity. In the l-th
layer, the electric field and the magnetic field of the TE mode light beam with incident angle  i is
given by [R1]
El ( x, y )  [ Al e  k lix ( x  xl ) eiklrx ( x  xl )  Bl e k lix ( x  xl ) e  iklrx ( x  xl ) ]e  k liy y eiklry y e z ,

(1)
1

 k y ik y
 k lix ( x  x l ) iklrx ( x  x l )
e
 Bl e k lix ( x  xl ) e  iklrx ( x  xl ) ]e liy e lry ,
Hl ( x, y )    klye x  klxe y [ Al e
0 l

where k l  k lr  ik li is the wave vector of the THz wave, (e x , e y , e z ) is the unit vectors in the
(x,y,z)-direction respectively,
and xl is the position of the l-th layer in the x-direction,
the circular frequency of the light beams,  0 is the permeability of vacuum,
 is
l  1 is the
permeability of the l-th layer. And the electric field and magnetic field of the TM mode light beam
with incident angle  i is given by
Hl ( x, y )  [ Al e  k lix ( x  xl ) eiklrx ( x  xl )  Bl ek lix ( x  xl ) e iklrx ( x  xl ) ]e  k liy y eiklry y e z ,

(2)
1

 k y ik y
 k lix ( x  x l ) iklrx ( x  x l )
e
 Bl ek lix ( x  xl ) e  iklrx ( x  xl ) ]e liy e lry ,
El ( x, y )    klye x  klxe y [ Al e
0 l

the electric fields of TE mode or the magnetic fields of TM mode in the (l+1)-th and l-th layer are
related by the matrix utilizing the boundary condition
  l   l 1  k lix d l iklx d l

e
e
 Al 1   2 l 1

  
 Bl 1    l 1   l e  k lix d l eiklx d l
 2
l 1

where  l 
klrx  ik lix
0 l
( l 
 l 1   l k
e
2 l 1

e  iklx d l 
l
l
 Al   t11 t12  Al 


 , (3)

l
l 
t22
 l   l 1 k lix d l  iklx d l  Bl   t21
 Bl 
e e

2 l 1

klrx  ik lix
 0 l
lix d l
) for TE (TM) mode, d l is the thickness of the l-th
layer. The light in the l-th layer is related to the incident fields by the transfer matrix
 Al   t11l 1 t12l 1   t110 t120  A0   T11 T12  A0 
   
    l 1 l 1  0
 .
0 
B
t
t
t
t
 l   21
22 
 21 22  B0   T21 T22  B0 
(4)
Thus, we can obtain the absorptance of the l-th layer to (l+m)-th layer  a using the Poynting
vector S  E  H
 a  [S( l 1)i  S( l  m 1)i  S( l 1)o  S( l  m 1)o ] / S0i ,
(5)
where S( l 1) i and S ( l 1) o [ S ( l  m 1) i and S ( l  m 1) o ] are the incident and outgoing Poynting
vectors in (l-1)-th and [(l+m+1)-th] layers, respectively, and S 0 i is the incident Poynting vector
in air.
S2. Distributed Bragg reflector with parallel layers
Fig. S1. Absorptance of shoulder-to-shoulder GM-PV subcells with different
thickness planar-shaped DBR (inset: schematic of shoulder-to-shoulder
GM-PV subcells).
To expand the photonic band gap of the distributed Bragg reflector (DBR), wedge-shaped
DBR is used in the calculation. Usually, the DBR is fabricated with planar-shaped layers. Limited
to the photonic band gap, perfect absorption is difficult to achieve in a wide wavelength range. But
by using shoulder-to-shoulder subcells with different DBR, nearly perfect absorption can be
achieved over a broad spectrum range [Fig. S1].
S3. The influence of the fabrication tolerances of the photonic crystal.
Fig. S2. Absorptance of GM-PV cell when a random thickness variation is
introduced in each layer in DBR, standard deviation (a)   1 nm, (b)   2
nm, and (c)   3 nm
The thickness variations of the layers in DBR cause the reflectivity and the resonant
frequency of the cavity may change as well. Similar to the actual fabrication process, random
thickness variation is introduced in each layer in DBR. The thickness variation d l follows a
2
normal distribution f   
exp[  2 ] , where parameter  is the standard
2
2  
1
deviation. The normal distribution is simulated by using the Box–Muller transform [R2]. The
calculated results are illustrated in Fig. S2. In each Figure, the absorptance is calculated by 60
times with random thickness variation.
For
  1 nm, more than 91.7% of the results show
that the average absorptance is larger than 0.9. For
  2 nm (   3 nm), more than 55%
(43%) percent of the results show that the average absorptance is larger than 0.9 and more than
70% (62%) percent of the results show that the average absorptance is larger than 0.8. For
 2
nm (   3 nm), the lowest average absorptance is only about 0.65 (0.35), which is caused by the
large thickness variation (>5 nm) in the layer most close to the defect layer.
S4. The outermost layer correction.
Fig. S3. Absorptance of GM-PV cell with defect layer thicknesses variation d c
and outermost layer thicknesses correction d ol . The inset shows the perfect
matched correction (blue dash-dotted line) and wedge-shaped correction (red
dotted line).
As an interference effect, the cavity resonant frequency depends not only on the optical path
length of the defect layers but also on the total optical path length of the DBR. Thus, by adding a
thickness correction d ol in the outermost layer, the resonant frequency of the cavity can be
adjusted. The reflectivity of the upper DBR can also be adjusted by varying the outermost layer
thickness, which will lead to variation of the Q-value of the microcavity.
Numerical results are
shown in Fig. S3. By adjusting the defect layer thickness, a resonance-induced absorption can be
achieved (i.e., d c  0 , dol  0 , black sold line). If the defect layer thickness reduces d c ,
the resonance-induced perfect absorption of the light incident on the cavity can no longer be
achieved. The absorptance is reduced ( d c  1 nm or d c  2 nm, dol  0 , green dashed
line). However, when introducing the outermost layer perfect matched correction (bule
dash-dotted)or wedge-shaped correction (red dotted line), the absorption can be enhanced.
Especially, a maximum absorption around
  580 nm can be achieved even with d c  1
nm by introducing the correction. It is because that the Q-value of the microcavity can be adjusted
by varying the thickness of the outermost layer. Therefore, even if the manufacturing deviation
occurs, by adjusting the thickness of the outermost layer it is possible to improve the quality and
reduce the existing number of discarded products.
S5. The tolerances of the photovoltaic cells position.
When the x position of the GM-PV cell has a certain deviation, the absorptance of GM-PV
cell decreases due to the lack of perfect matching. This variation is equivalent to the variation in
the thickness of the defect layer. The slope of the defect layer thickness variation is approximately
80nm / 60mm  1.33  10 6 . The variation of 1.5 mm in the x position means that the defect
layer thickness exhibits a change of 2 nm, i.e., the deviation of x position impose significant
effects on the absorptance of GM-PV cell. Certainly, with more efficient spectrum-splitting system,
the slope of thickness variation can be reduced and thus the effects induced by the deviation of
x-position can be weakened. Additionally, the adjustment can be implemented with a feedback
mechanical system.
The deviation of the GM-PV cell z-position is equivalent to the change of spot size. Its effect
is equivalent to that of aberration. Smaller diameter of the lens and longer focal length means less
effect induced by the deviation of z-position. When the diameter is 10 cm and the focal length is
50 cm, if the deviation of z-postion is 0.5 cm, the diameter of the light spot will increase by about
1 mm, smaller than the variation of light spot induced by aberration, i.e., the effect is little.
S6. The external quantum efficiency of the proposed GM-PV cell.
Finally, we discuss the external quantum efficiency of the proposed GM-PV cell. As shown
in the main tex of the paper, the GM-PV cell exhibits up to >90% light absorptance in the 450-670
nm wavelength range. By choose appropriate materials of DBR and spectrum-splitting structure,
Fig. S4. Absorption of the GM-PV cell in this paper (red dashed line) and
in Ref. [R3] (black solid line). The inset shows the AM1.5G spectrum[R4]
broadband perfect absorption can be achieved in <450 nm wavelength range. Thus, >25% of the
total power spectrum from the sun can be absorbed by the proposed GM-PV cell. The external
quantum efficiency can be larger than 20%. To get perfect absorption with wavelength >700 nm,
shoulder-to-shoulder cells consisting of narrower band gap graphene-MoSe2 (1.5 eV) and
graphene-MoTe2 (1.1 eV) [R5] should be added [R6]. The estimated external quantum efficiency
can be larger than 80%. As a comparison, we also show the absorption of the GM-PV cell in Ref.
[R3] in Fig. S4. By using the chirped-planar-dielectric cavities, the absorption of GM-PV cell can
absorb as much as 33% of incident visible light over a 300 nm bandwidth, and the external
quantum efficiency of GM-PV cell can be enhanced 3.6 times to a predicted value of 7.09%.
Referrence
[R1] Born and E. Wolf, Principles of Optics. Pergamon, Oxford, U.K., (1989), pp. 38–74.
[R2] G. E. P. Box and Mervin E. Muller, A Note on the Generation of Random Normal
Deviates, The Annals of Mathematical Statistics (1958), Vol. 29, No. 2 pp. 610–611.
[R3]The
AM1.5G
spectrum
was
taken
from
the
NREL
website:
http://rredc.nrel.gov/solar/spectra/am1.5 and integrated with the trapezoid rule.
[R5] Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N. & Strano, M. S. Electronics
and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotech. 7,
699–712 (2012).
[R6]Liu, J.-T., Deng, X.-H., Yang, W. & Li, J. Perfect light trapping in nanoscale thickness
semiconductor films with a resonant back reflector and spectrum-splitting structures. Phys. Chem.
Chem. Phys. 17, 3303 (2015).